A uniform thin disk rolls without slipping on a plane and a force is being applied at its center parallel to the plane. Find the lagrangian and thereby the generalized force.
The Attempt at a Solution
This is my first ever exercise of this kind. We first note that U=0 so the lagrangian is simply L=T. We then want to express the kinetic energy T in terms of coordinates, which contain the constraints of the motion implicitly. Therefore we should use polar coordinates. Correct so far?
We then get:
L = T = ½m([itex]\omega[/itex]2r2) (1)
And now lagranges equation says:
d/dt[dT/dqj'] - dT/dqj = Qj
where Qj is the generalized force. There is for this motion one equation of the above kind - one for theta and one for r.
Should I now just differentiate with respect to r and [itex]\theta[/itex] and make two separate equations of the above kind of which I can find the components of the generalized force?
I just don't get anything very sensible when I differentiate the expression for L above with the two variables, and when my teacher did it I think he just differentiated with respect to x - why is that? Shouldn't you use the generalized coordinates?